Explicit port-Hamiltonian FEM models for geometrically nonlinear mechanical systems

In this article, we present the port-Hamiltonian representation, the structure preserving discretization and the resulting finite-dimensional state space model of geometrically nonlinear mechanical systems based on a mixed finite element formulation. This article focuses on St. Venant-Kirchhoff materials connecting the Green strain and the second Piola-Kirchhoff stress tensor in a linear relationship which allows a port-Hamiltonian representation by means of its co-energy (effort) variables. Due to treatment of both Dirichlet and Neumann boundary conditions in the appropriate variational formulation, the resulting port-Hamiltonian state space model features both of them as explicit (control) inputs. Numerical experiments generated with FEniCS illustrate the properties of the resulting FE models.